Abstract
A new approach for optimization of control problems defined by fully implicit differential-algebraic equations is described in the paper. The main feature of the approach is that system equations are substituted by discrete-time implicit equations resulting from the integration of the system equations by an implicit Runge–Kutta method. The optimization variables are parameters of piecewise constant approximations to control functions; thus, the control problem is reduced to the control space only. The method copes efficiently with problems defined by large-scale differential-algebraic equations.
Similar content being viewed by others
References
Brenan, K. E., Campbell, S. L., and Petzold, L.R., Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North-Holland, New York, New York, 1989.
Pytlak, R., and Vinter, R. B., A Feasible Directions Type Algorithm for Optimal Control Problems with State and Control Constraints: Convergence Analysis, Research Report C96-24, Centre for Process Systems Engineering, Imperial College, London, England, SIAM Journal of Optimization (to appear).
Pytlak, R., Second-Order Method for Optimal Control Problems with Piecewise Constant Controls and State Constraints, Proceedings of the 34th IEEE Conference on Decision and Control, New Orleans, Louisiana, pp. 625–630, 1995.
Tanartkit, P., and Biegler, L. T., Stable Decomposition for Dynamic Optimization, Industrial and Engineering Chemistry Research, Vol. 34, pp. 1253–1266, 1995.
Biegler, L. T., Nocedal, J., and Schmid, C., A Reduced Hessian Method for Large-Scale Constrained Optimization, SIAM Journal on Optimization, Vol. 5, pp. 314–347, 1995.
Mohideen, M. J., Perkins, J. D., and Pistikopoulos, E. N., Optimal Design of Dynamic Systems under Uncertainty, AIChE Journal, Vol. 42, pp. 2251–2272, 1995.
Polak, E., Computational Methods in Optimization: Unified Approach, Academic Press, New York, New York, 1971.
Duff, I. S., Erisman, A. M., and Reid, J. K., Direct Methods for Sparse Matrices, Oxford Science Publications, Oxford, England, 1989.
Hairer, E., Lubich, C., and Roche, M., The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, Lecture Notes in Mathematics, Springer Verlag, Berlin, Germany, Vol. 1409, 1989.
Hairer, E., and Wanner, G., Solving Ordinary Differential Equations, Vol. 2: Stiff and Differential-Algebraic Problems, Springer Verlag, Berlin Germany, 1991.
Pytlak, R., A Variational Approach to the Discrete Maximum Principle, IMA Journal of Mathematical Control and Information, Vol. 9, pp. 197–220, 1992.
Bertsekas, D. P., Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York, New York, 1982.
Fedorenko, R. P., Priblizhyonnoye Reshenyie Zadach Optimalnovo Upravlenya, Nauka, Moscow, Russia, 1978 (in Russian).
Pytlak, R., and Malinowski, K., Optimal Scheduling of Reservoir Releases during Flood: Deterministic Optimization Problems, Parts 1 and 2, Journal of Optimization Theory and Applications, Vol. 61, pp. 409–449, 1989.
Mayne, D. Q., and Polak, E., An Exact Penalty Function Algorithm for Control Problems with State and Control Constraints, IEEE Transactions on Automatic Control, Vol. 32, pp. 380–387, 1987.
Shampine, L. F., Implementation of Implicit Formulas for the Solution of ODEs, SIAM Journal on Scientific and Statistical Computing, Vol. 1, pp. 103–118, 1980.
Petzold, L., and LÖtstedt, P., Numerical Solution of Nonlinear Differential Equations with Algebraic Constraints, II: Practical Implications, SIAM Journal on Scientific and Statistical Computing, Vol. 7, pp. 720–733, 1986.
ANONYMOUS, Harwell Subroutine Library Specification, pp. 261–273, 1993.
Pytlak, R., Mohideen, M. J., and Pistikopoulos, E. N., Numerical Procedure for Optimal Control of Differential-Algebraic Equations, Research Report C96-23, Centre for Process Systems Engineering, Imperial College, London, England, 1996.
Barton, P. J., The Modelling and Simulation of Combined Discrete/Continuous Processes, PhD Thesis, University of London, London, England, 1992.
Vassiliadis, V., Computational Solution of Dynamic Optimization Problems with General Differential-Algebraic Constraints, PhD Thesis, University of London, London, England, 1993.
Brooke, A., Kendrick, D., and Meeraus, M., GAMS: A User's Guide, Scientific Press, Redwood City, California, 1988.
Drud, A. S., A GRG Code for Large Sparse Dynamic Nonlinear Optimization Problems, Mathematical Programming, Vol. 31, pp. 153–191, 1985.
Abadie, J., and Carpentier, J., Generalization of the Wolfe Reduced Gradient Method to the Case of Nonlinear Constraints, Optimization, R. Fletcher, ed., Academic Press, London, pp. 37–49, 1969.
Wright, S. J., Structured Interior-Point Methods for Optimal Control, Proceedings of the 30th IEEE Conference on Decision and Control, Brighton, England, pp. 1711–1716, 1991.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pytlak, R. Runge–Kutta Based Procedure for the Optimal Control of Differential-Algebraic Equations. Journal of Optimization Theory and Applications 97, 675–705 (1998). https://doi.org/10.1023/A:1022698311155
Issue Date:
DOI: https://doi.org/10.1023/A:1022698311155